The lowest common multiple is the first number both counts land on
Where two counts first meet
Count in fours: 4, 8, 12, 16, 20, 24. Then count in sixes: 6, 12, 18, 24. The two counts first land on the same number at 12.
The smallest number that is a multiple of both of two numbers is called their Lowest Common Multiple, or LCM for short. The LCM of 4 and 6 is 12.
Counting in fours lands on 4, 8, 12, 16, 20 and 24.
On the same scale, counting in sixes lands on 6, 12, 18 and 24. 12 is the first landing the two counts share.
The first meeting, not a later one
24 is on both counts as well, but it comes later. The LCM is the lowest common multiple, so it is 12, not 24.
Every other common multiple of 4 and 6 is a multiple of the LCM: 24 = 12 × 2 and 36 = 12 × 3. After the first meeting, the two counts meet again every 12.
The LCM from the primes
Write both numbers as primes: 4 = 2 × 2 and 6 = 2 × 3. A multiple of 4 has 2 × 2 inside it, and a multiple of 6 has 2 × 3 inside it. The smallest number with both inside it is 2 × 2 × 3 = 12. Its two 2s are what 4 needs, and one of those 2s with the 3 is what 6 needs.
So take every prime that appears in either list, as many times as the list that has it most. 2 appears twice in 4 and once in 6, so take it twice. 3 appears only in 6, once, so take it once. 2 × 2 × 3 = 12.
4 = 2 × 2 and 6 = 2 × 3 share one 2, which goes in the overlap once. Everything in the diagram multiplied together, 2 × 2 × 3 = 12, is the LCM.
Bigger numbers
For 24 and 36: 24 = 2 × 2 × 2 × 3 and 36 = 2 × 2 × 3 × 3. 2 appears three times in 24 and twice in 36, so take it three times. 3 appears once in 24 and twice in 36, so take it twice. The LCM is 2 × 2 × 2 × 3 × 3 = 72.
The primes the two lists share, 2 × 2 × 3 = 12, make the HCF. The HCF takes what both lists have; the LCM takes everything either list has.
Slide the shared factors into the overlap
Slide the shared primes of 24 and 36 into the overlap, one at a time. The overlap multiplies to the HCF, 12. Everything in the diagram, with each shared prime counted once, multiplies to the LCM, 72.
A check
The HCF multiplied by the LCM always equals the two numbers multiplied. For 24 and 36: 12 × 72 = 864, and 24 × 36 = 864. For 4 and 6: the HCF is 2, and 2 × 12 = 24 = 4 × 6.
Multiplying can overshoot
Multiplying the two numbers gives a common multiple, but not always the lowest. 4 × 6 = 24 uses the shared 2 twice, once for 4 and once for 6, when the LCM needs it only once.
The product is the LCM only when the two numbers share no factor except 1. 8 and 9 share none, so their LCM is 8 × 9 = 72.
Adding is never the way: 4 + 6 = 10 is not a multiple of 4 or of 6. And the LCM is never smaller than the larger number, because the larger number divides it.
HCF and LCM · LCM: The Least Spent on Equal Numbers of Buns and Sausages
QuestionAt a supermarket, hot dog buns are sold in packs of 6 for $4, and sausages are sold in packs of 8 for $5. Melissa wants to buy exactly the same number of buns and sausages, spending as little as she can. (a) How many buns and how many sausages does she buy? (b) What is the least amount of money she can spend?
1.Buns come in packs of 6, so she can buy 6, 12, 18, 24 or 30 buns, and so on. These are the multiples of 6.
Buns come in packs of 6: she can buy 6, 12, 18, 24 or 30 buns. 2.Sausages come in packs of 8, so she can buy 8, 16, 24 or 32 sausages, and so on. These are the multiples of 8.
Sausages come in packs of 8: she can buy 8, 16, 24 or 32 sausages. 3.(a) The first number on both lists is 24, the LCM of 6 and 8. She buys 24 buns and 24 sausages.
(a) The first number on both lists is 24, the LCM of 6 and 8. 4.24 buns is 24 ÷ 6 = 4 packs, which cost 4 × $4 = $16. 24 sausages is 24 ÷ 8 = 3 packs, which cost 3 × $5 = $15.
24 buns is 4 packs: 4 × $4 = $16. 24 sausages is 3 packs: 3 × $5 = $15. 5.(b) The least she can spend is $16 + $15 = $31. The next number on both lists, 48, would need twice as many packs and cost twice as much.
(b) The least she can spend is $16 + $15 = $31.
Answer: (a) 24 buns and 24 sausages; (b) $31
Watch out for
- Finding the LCM of the prices, 4 and 5, and answering $20. The buns and the sausages must match in number, so the LCM comes from the pack sizes, 6 and 8, not from the prices.
- Multiplying 24 by the price of a pack, 24 × $4 = $96. $4 buys a pack of 6 buns, not one bun, so first find the number of packs: 24 ÷ 6 = 4.
HCF and LCM · LCM: The Next Time Three Lighthouses Flash Together
QuestionThree lighthouses flash their lights. Lighthouse A flashes once every 24 seconds, Lighthouse B once every 36 seconds and Lighthouse C once every 60 seconds. All three flash together at 8:00 p.m. (a) At what time will they next flash together? (b) How many times will all three flash together between 8:01 p.m. and 8:35 p.m.?
1.The lighthouses flash together again after a number of seconds that 24, 36 and 60 all divide into: a common multiple. The first such time is their LCM.
The lighthouses flash together again at a number of seconds that 24, 36 and 60 all divide into. 2.24, 36 and 60 are all multiples of 12: 24 = 12 × 2, 36 = 12 × 3 and 60 = 12 × 5. No number except 1 divides any two of 2, 3 and 5.
24, 36 and 60 are all multiples of 12: 12 × 2, 12 × 3 and 12 × 5. No number except 1 divides any two of 2, 3 and 5. 3.So the LCM is 12 times the smallest number that 2, 3 and 5 all divide into, which is 2 × 3 × 5 = 30. The LCM is 12 × 30 = 360 seconds, which is 360 ÷ 60 = 6 minutes.
The LCM is 12 × 2 × 3 × 5 = 360 seconds, which is 6 minutes. 4.(a) They next flash together 6 minutes after 8:00 p.m., at 8:06 p.m.
(a) They next flash together at 8:06 p.m., and then every 6 minutes. 5.After that they flash together every 6 minutes: at 8:06, 8:12, 8:18, 8:24 and 8:30 p.m. The next time, 8:36 p.m., is after 8:35 p.m.
Between 8:01 and 8:35 p.m. they flash together at 8:06, 8:12, 8:18, 8:24 and 8:30. 8:36 is too late. 6.(b) All three flash together 5 times between 8:01 p.m. and 8:35 p.m.
(b) All three flash together 5 times between 8:01 p.m. and 8:35 p.m.
Answer: (a) 8:06 p.m.; (b) 5 times
Watch out for
- Using the HCF of 24, 36 and 60, which is 12 seconds. None of the lighthouses flashes 12 seconds after 8:00 p.m., because 12 is shorter than every one of the times. The time must be a multiple of 24, 36 and 60, so it is the LCM.
- Counting the flash at 8:00 p.m. or the one at 8:36 p.m. and answering 6 or 7. 8:00 p.m. is before 8:01 p.m., and 8:36 p.m. is after 8:35 p.m., so neither is between them.
Your turn
1.Lowest common multiple of 4 and 10?